WASHINGTON — The tradition-laden Senate voted Thursday to modestly curb filibusters, using a bipartisan consensus rare in today’s hyper-partisan climate to make it a bit harder but not impossible for outnumbered senators to sink bills and nominations.

The rules changes would reduce yet not eliminate the number of times opponents — usually minority Republicans these days — can use filibusters, procedural tactics that can derail legislation and that can be stopped only by the votes of 60 of the 100 senators.

In return, the majority — Democrats today — would have to allow two minority amendments on bills, a response to Republican complaints that Democrats often prevent them from offering any amendments at all. The new procedures also would limit the time spent debating some bills and nominations, allowing some to be completed in hours that could otherwise take a day or more.

The changes were broken into two pieces and approved by votes of 78-16 and 86-9.

The two votes and a brief debate took less than an hour, impressively quick for the Senate. They came after a more typical day that featured a sprinkling of senators’ speeches and long periods when the Senate chamber idled with no one talking, while private negotiations off the floor nailed down final details.

President Obama said the Senate action would help his agenda in Congress.

‘‘At a time when we face critical decisions on a whole range of issues — from preventing further gun violence, to reforming our broken immigration system, to getting our fiscal house in order and creating good paying jobs — we cannot afford unnecessary obstruction,’’ Obama said in a written statement.

The pact leaves the Senate’s minority party with far more power than it has in the House, where rules let a united majority party easily muscle through its priorities.

It also falls short of changes majority leader Harry Reid, a Nevada Democrat, had been threatening to ram through using the 55 votes Democrats have, a technique nicknamed the ‘‘nuclear option’’ because it is considered likely to produce harsh GOP retaliation that could grind work to a virtual halt.

Senator Elizabeth Warren said she is disappointed Senate leaders were not able to come up with tougher changes to the filibuster rule that allows minority parties to bottle up legislation.

The Massachusetts Democrat had vowed to join other Democrats to limit use of the filibuster — a move she said would help break Washington gridlock by curbing the ability of Republicans to block votes.

Warren had said senators who want to filibuster should be required to stand on the Senate floor and speak continuously rather than merely threatening to do so.

She said in a statement Thursday that ‘‘some change is better than no change at all.’’

The deal gave each side something it wanted: Quicker action for Democrats, guaranteed amendments for Republicans. And it avoided clamping tight limits on filibusters that would alienate veteran lawmakers wary that their party could fall into the minority after any election.

Months ago, Reid said he favored completely banning filibusters when the Senate tries to begin debating a measure, a tactic Republicans have been using more in recent years. He threatened to use Democrats’ strength in the Senate to enact that change and perhaps others by a simple majority vote, instead of the two-thirds majority most rules changes require.

Senator John McCain, and Arizona Republican, among a bipartisan group of eight senators who proposed changes similar to those embraced in the deal, said the agreement showed that ‘‘it’s possible for a group of us to join together on very difficult issues.’’

The liberal group Common Cause, which has filed a lawsuit challenging the constitutionality of the filibuster, criticized Reid for the deal, saying the senator ‘‘has gone missing in the fight for filibuster reform.’’

Senate will vote Monday on Hurricane Sandy relief bill

WASHINGTON — The Senate is set to vote Monday on a $50.5 billion emergency aid bill to help victims of Hurricane Sandy.

Northeast lawmakers from both parties hope to win Senate approval of the measure and send it to President Obama, who has said he would sign it. The House passed it last week.